The length of the hypotenuse is 13. For ∠S, the length of the opposite side is 5, and the length of the adjacent side is 12.

Sin S = opp. / hyp. = 5 / 13 ≈ 0.3846

Cos S = adj. / hyp. = 12 / 13 ≈ 0.9231

Tan S = opp. / adj. = 5 / 12 ≈ 0.4167

Solution (b) :

The length of the hypotenuse is 13. For ∠R, the length of the opposite side is 12, and the length of the adjacent side is 5.

Sin R = opp. / hyp. = 12 / 13 ≈ 0.9231

Cos R = adj. / hyp. = 5 / 13 ≈ 0.3846

Tan R = opp. / adj. = 12 / 5 ≈ 2.4

Example 3 :

Find the sine, the cosine, and the tangent of 45°.

Solution :

Begin by sketching a 45°-45°-90° triangle. Because all such triangles are similar, we can make calculations simple by choosing 1 as the length of each leg. From the Pythagorean Theorem, it follows that the length of the hypotenuse is √2.

Sin 45° = opp. / hyp. = 1 / √2 = √2 / 2≈ 0.7071

Cos 45° = adj. / hyp. = 1 / √2 = √2 / 2 ≈ 0.7071

Tan 45° = opp. / adj. = 1 / 1 = 1

Example 4 :

Find the sine, the cosine, and the tangent of 30°.

Solution :

Begin by sketching a 30°-60°-90° triangle. To make the calculations simple, we can choose 1 as the length of the shorter leg. From 30°-60°-90° Triangle theorem, it follows that the length of the longer leg is √3 and the hypotenuse is 2.

Sin 30° = opp. / hyp. = 1 / 2 = 0.5

Cos 30° = adj. / hyp. = √3 / 2 ≈ 0.8660

Tan 30° = opp. / adj. = 1 / √3 = √3 / 3 ≈ 0.5774

Example 5 :

Jacob is measuring the height of a Sitka spruce tree in North Carolina. He stands 45 feet from the base of the tree. He measures the angle of elevation from a point on the ground to the top of the tree to be 59°. How can he estimate the height of the tree ?

Solution :

To estimate the height of the tree, Jacob can write a trigonometric ratio that involves the height h and the known length of 45 feet.

In the above right triangle, for the angle 59°, h is opposite side and the side has length 45 ft is adjacent side.

The trigonometric ratio that involves opposite side and adjacent side is tangent.

Write ratio :

tan 59° = opp. / adj.

Substitute.

tan 59° = h / 45

Multiply each side by 45.

45 ⋅ tan 59° = h

Use calculator or table to find the value of tan 59° and substitute.

45 ⋅ 1.6643 = h

Simplify.

74.9 ≈ h

So, the tree is about 75 feet tall.

Example 6 :

The escalator at the University Metro Rail Station near University of Miami in Coral Gables, Florida rises 76 feet at a 30° angle as shown in the diagram below. Find the distance d a person travels on the escalator stairs.

Solution :

To find the distance d a person travels on the escalator stairs, we can write a trigonometric ratio that involves the side has length d and the known length of 76 feet.

In the above right triangle, for the angle 30°, d is hypotenuse and the side has length 76 feet is opposite side.

The trigonometric ratio that involves opposite side and hypotenuse is sin.

Write ratio :

sin 30° = opp. / hyp.

Substitute.

sin 30° = 76 / d

Multiply each side by d.

d ⋅ sin 30° = 76

Substitute 0.5 for sin 30°.

d ⋅ 0.5 = 76

Divide each side by 0.5

d = 76 / 0.5

d = 152

So, the person travels 152 feet on the escalator stairs.

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